On Prime Numbers and Related Applications
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On the First Occurrences of Gaps Between Primes in a Residue Class
On the First Occurrences of Gaps Between Primes in a Residue Class Alexei Kourbatov JavaScripter.net Redmond, WA 98052 USA [email protected] Marek Wolf Faculty of Mathematics and Natural Sciences Cardinal Stefan Wyszy´nski University Warsaw, PL-01-938 Poland [email protected] To the memory of Professor Thomas R. Nicely (1943–2019) Abstract We study the first occurrences of gaps between primes in the arithmetic progression (P): r, r + q, r + 2q, r + 3q,..., where q and r are coprime integers, q > r 1. ≥ The growth trend and distribution of the first-occurrence gap sizes are similar to those arXiv:2002.02115v4 [math.NT] 20 Oct 2020 of maximal gaps between primes in (P). The histograms of first-occurrence gap sizes, after appropriate rescaling, are well approximated by the Gumbel extreme value dis- tribution. Computations suggest that first-occurrence gaps are much more numerous than maximal gaps: there are O(log2 x) first-occurrence gaps between primes in (P) below x, while the number of maximal gaps is only O(log x). We explore the con- nection between the asymptotic density of gaps of a given size and the corresponding generalization of Brun’s constant. For the first occurrence of gap d in (P), we expect the end-of-gap prime p √d exp( d/ϕ(q)) infinitely often. Finally, we study the gap ≍ size as a function of its index in thep sequence of first-occurrence gaps. 1 1 Introduction Let pn be the n-th prime number, and consider the difference between successive primes, called a prime gap: pn+1 pn. -
Ramanujan, Robin, Highly Composite Numbers, and the Riemann Hypothesis
Contemporary Mathematics Volume 627, 2014 http://dx.doi.org/10.1090/conm/627/12539 Ramanujan, Robin, highly composite numbers, and the Riemann Hypothesis Jean-Louis Nicolas and Jonathan Sondow Abstract. We provide an historical account of equivalent conditions for the Riemann Hypothesis arising from the work of Ramanujan and, later, Guy Robin on generalized highly composite numbers. The first part of the paper is on the mathematical background of our subject. The second part is on its history, which includes several surprises. 1. Mathematical Background Definition. The sum-of-divisors function σ is defined by 1 σ(n):= d = n . d d|n d|n In 1913, Gr¨onwall found the maximal order of σ. Gr¨onwall’s Theorem [8]. The function σ(n) G(n):= (n>1) n log log n satisfies lim sup G(n)=eγ =1.78107 ... , n→∞ where 1 1 γ := lim 1+ + ···+ − log n =0.57721 ... n→∞ 2 n is the Euler-Mascheroni constant. Gr¨onwall’s proof uses: Mertens’s Theorem [10]. If p denotes a prime, then − 1 1 1 lim 1 − = eγ . x→∞ log x p p≤x 2010 Mathematics Subject Classification. Primary 01A60, 11M26, 11A25. Key words and phrases. Riemann Hypothesis, Ramanujan’s Theorem, Robin’s Theorem, sum-of-divisors function, highly composite number, superabundant, colossally abundant, Euler phi function. ©2014 American Mathematical Society 145 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 146 JEAN-LOUIS NICOLAS AND JONATHAN SONDOW Figure 1. Thomas Hakon GRONWALL¨ (1877–1932) Figure 2. Franz MERTENS (1840–1927) Nowwecometo: Ramanujan’s Theorem [2, 15, 16]. -
A NOTE on the GAUSSIAN MOAT PROBLEM Real Axis
A NOTE ON THE GAUSSIAN MOAT PROBLEM MADHUPARNA DAS Abstract. The Gaussian moat problem asks whether it is possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded. In this paper, we have proved that the answer is ‘No’, that is an infinite sequence of distinct Gaussian prime numbers can not be bounded by an absolute constant, for the Gaussian primes p = a2 + b2 with a, b =6 0. We consider each prime (a, b) as a lattice point on the complex plane and use their properties to prove the main result. 1. Introduction Primes are always interesting topics for mathematicians. After lots of research work still, we don’t have any explicit formula for the distribution of the natural prime numbers. Prime number theorem gives an asymptotic formula for primes with some bounded error. Also, the prime number theorem implies that there are arbitrarily large gaps in the sequence of prime numbers, and this can also be proved directly: for any n, the n 1 consecutive numbers n!+2,n!+3,...,n!+ n are all composite. It’s very easy for natural primes− that it is not possible to walk to infinity with stepping on the natural prime numbers with bounded length gap. Now we can think about the same problem for the complex prime numbers or explicitly for ‘Gaussian Primes’. So, the question says: “Whether it is possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded. -
ON the PRIMALITY of N! ± 1 and 2 × 3 × 5 ×···× P
MATHEMATICS OF COMPUTATION Volume 71, Number 237, Pages 441{448 S 0025-5718(01)01315-1 Article electronically published on May 11, 2001 ON THE PRIMALITY OF n! 1 AND 2 × 3 × 5 ×···×p 1 CHRIS K. CALDWELL AND YVES GALLOT Abstract. For each prime p,letp# be the product of the primes less than or equal to p. We have greatly extended the range for which the primality of n! 1andp# 1 are known and have found two new primes of the first form (6380! + 1; 6917! − 1) and one of the second (42209# + 1). We supply heuristic estimates on the expected number of such primes and compare these estimates to the number actually found. 1. Introduction For each prime p,letp# be the product of the primes less than or equal to p. About 350 BC Euclid proved that there are infinitely many primes by first assuming they are only finitely many, say 2; 3;:::;p, and then considering the factorization of p#+1: Since then amateurs have expected many (if not all) of the values of p# 1 and n! 1 to be prime. Careful checks over the last half-century have turned up relatively few such primes [5, 7, 13, 14, 19, 25, 32, 33]. Using a program written by the second author, we greatly extended the previous search limits [8] from n ≤ 4580 for n! 1ton ≤ 10000, and from p ≤ 35000 for p# 1top ≤ 120000. This search took over a year of CPU time and has yielded three new primes: 6380!+1, 6917!−1 and 42209# + 1. -
Sum of Divisors of the Primorial and Sum of Squarefree Parts
International Mathematical Forum, Vol. 12, 2017, no. 7, 331 - 338 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.7113 Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts Rafael Jakimczuk Divisi´onMatem´atica,Universidad Nacional de Luj´an Buenos Aires, Argentina Copyright c 2017 Rafael Jakimczuk. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduc- tion in any medium, provided the original work is properly cited. Abstract Let pn be the n-th prime and σ(n) the sum of the positive divisors of n. Let us consider the primorial Pn = p1:p2 : : : pn, the sum of its Qn positive divisors is σ(Pn) = i=1(1 + pi). In the first section we prove the following asymptotic formula n 6 σ(P ) = Y(1 + p ) = eγ P log p + O (P ) : n i π2 n n n i=1 Let a(k) be the squarefree part of k, in the second section we prove the formula π2 X a(k) = x2 + o(x2): 30 1≤k≤x We also study integers with restricted squarefree parts and generalize these results to s-th free parts. Mathematics Subject Classification: 11A99, 11B99 Keywords: Primorial, divisors, squarefree parts, s-th free parts, average of arithmetical functions 332 Rafael Jakimczuk 1 Sum of Divisors of the Primorial In this section p denotes a positive prime and pn denotes the n-th prime. The following Mertens's formulae are well-known (see [5, Chapter VI]) 1 1 ! X = log log x + M + O ; (1) p≤x p log x where M is called Mertens's constant. -
Various Arithmetic Functions and Their Applications
University of New Mexico UNM Digital Repository Mathematics and Statistics Faculty and Staff Publications Academic Department Resources 2016 Various Arithmetic Functions and their Applications Florentin Smarandache University of New Mexico, [email protected] Octavian Cira Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebra Commons, Applied Mathematics Commons, Logic and Foundations Commons, Number Theory Commons, and the Set Theory Commons Recommended Citation Smarandache, Florentin and Octavian Cira. "Various Arithmetic Functions and their Applications." (2016). https://digitalrepository.unm.edu/math_fsp/256 This Book is brought to you for free and open access by the Academic Department Resources at UNM Digital Repository. It has been accepted for inclusion in Mathematics and Statistics Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. Octavian Cira Florentin Smarandache Octavian Cira and Florentin Smarandache Various Arithmetic Functions and their Applications Peer reviewers: Nassim Abbas, Youcef Chibani, Bilal Hadjadji and Zayen Azzouz Omar Communicating and Intelligent System Engineering Laboratory, Faculty of Electronics and Computer Science University of Science and Technology Houari Boumediene 32, El Alia, Bab Ezzouar, 16111, Algiers, Algeria Octavian Cira Florentin Smarandache Various Arithmetic Functions and their Applications PONS asbl Bruxelles, 2016 © 2016 Octavian Cira, Florentin Smarandache & Pons. All rights reserved. This book is protected by copyright. No part of this book may be reproduced in any form or by any means, including photocopying or using any information storage and retrieval system without written permission from the copyright owners Pons asbl Quai du Batelage no. -
Prime Harmonics and Twin Prime Distribution
Prime Harmonics and Twin Prime Distribution Serge Dolgikh National Aviation University, Kyiv 02000 Ukraine, [email protected] Abstract: Distribution of twin primes is a long stand- in the form: 0; p−1;:::;2;1 with the value of 0 the highest ing problem in the number theory. As of present, it is in the cycle of length p. Trivially, the positions with the not known if the set of twin primes is finite, the problem same value of the modulo p are separated by the minimum known as the twin primes conjecture. An analysis of prime of p odd steps. modulo cycles, or prime harmonics in this work allowed For a prime p, the prime harmonic function hp(x) can be to define approaches in estimation of twin prime distri- defined on IN1 as the modulo of x by p in the above format: butions with good accuracy of approximation and estab- lish constraints on gaps between consecutive twin prime Definition 1. A single prime harmonic hp(x) is defined for pairs. With technical effort, the approach and the bounds an odd integer x ≥ 1 as: obtained in this work can prove sufficient to establish that hp(1) := (p − 1)=2 the next twin prime exists within the estimated distance, hp(n + 1) := hp(n) - 1, hp(n) > 0 leading to the conclusion that the set of twin primes is un- hp(n + 1) := p − 1, hp(n) = 0 limited and reducing the infinitely repeating distance be- Clearly, h (x) = 0 ≡ p j x. tween consecutive primes to two. -
Multiplication Modulo N Along the Primorials with Its Differences And
Multiplication Modulo n Along The Primorials With Its Differences And Variations Applied To The Study Of The Distributions Of Prime Number Gaps A.K.A. Introduction To The S Model Russell Letkeman r. letkeman@ gmail. com Dedicated to my son Panha May 19, 2013 Abstract The sequence of sets of Zn on multiplication where n is a primorial gives us a surprisingly simple and elegant tool to investigate many properties of the prime numbers and their distributions through analysis of their gaps. A natural reason to study multiplication on these boundaries is a construction exists which evolves these sets from one primorial boundary to the next, via the sieve of Eratosthenes, giving us Just In Time prime sieving. To this we add a parallel study of gap sets of various lengths and their evolution all of which together informs what we call the S model. We show by construction there exists for each prime number P a local finite probability distribution and it is surprisingly well behaved. That is we show the vacuum; ie the gaps, has deep structure. We use this framework to prove conjectured distributional properties of the prime numbers by Legendre, Hardy and Littlewood and others. We also demonstrate a novel proof of the Green-Tao theorem. Furthermore we prove the Riemann hypoth- esis and show the results are perhaps surprising. We go on to use the S model to predict novel structure within the prime gaps which leads to a new Chebyshev type bias we honorifically name the Chebyshev gap bias. We also probe deeper behavior of the distribution of prime numbers via ultra long scale oscillations about the scale of numbers known as Skewes numbers∗. -
Miscellaneous HOL Examples
Miscellaneous HOL Examples December 12, 2016 Contents 1 Some Isar command definitions 15 1.1 Diagnostic command: no state change............. 16 1.2 Old-style global theory declaration............... 16 1.3 Local theory specification.................... 16 2 Infinite Sets and Related Concepts 17 2.1 Infinitely Many and Almost All................. 18 2.2 Enumeration of an Infinite Set................. 21 3 Ad Hoc Overloading 24 3.1 Plain Ad Hoc Overloading.................... 24 3.2 Adhoc Overloading inside Locales................ 25 4 Permutation Types 27 5 Example of Declaring an Oracle 29 5.1 Oracle declaration........................ 29 5.2 Oracle as low-level rule...................... 29 5.3 Oracle as proof method..................... 30 6 Abstract Natural Numbers primitive recursion 34 7 Proof by guessing 37 8 Examples of function definitions 38 8.1 Very basic............................. 38 8.2 Currying.............................. 38 8.3 Nested recursion......................... 38 8.3.1 Here comes McCarthy's 91-function.......... 39 8.4 More general patterns...................... 39 8.4.1 Overlapping patterns................... 39 8.4.2 Guards.......................... 40 1 8.5 Mutual Recursion......................... 41 8.6 Definitions in local contexts................... 41 8.7 fun-cases ............................. 42 8.7.1 Predecessor........................ 42 8.7.2 List to option....................... 42 8.7.3 Boolean Functions.................... 43 8.7.4 Many parameters..................... 43 8.8 Partial Function Definitions................... 43 8.9 Regression tests.......................... 43 8.9.1 Context recursion.................... 44 8.9.2 A combination of context and nested recursion.... 44 8.9.3 Context, but no recursive call.............. 44 8.9.4 Tupled nested recursion................. 44 8.9.5 Let............................. 44 8.9.6 Abbreviations...................... -
Integer Sequences
UHX6PF65ITVK Book > Integer sequences Integer sequences Filesize: 5.04 MB Reviews A very wonderful book with lucid and perfect answers. It is probably the most incredible book i have study. Its been designed in an exceptionally simple way and is particularly just after i finished reading through this publication by which in fact transformed me, alter the way in my opinion. (Macey Schneider) DISCLAIMER | DMCA 4VUBA9SJ1UP6 PDF > Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular -
An Amazing Prime Heuristic
AN AMAZING PRIME HEURISTIC CHRIS K. CALDWELL 1. Introduction The record for the largest known twin prime is constantly changing. For example, in October of 2000, David Underbakke found the record primes: 83475759 · 264955 ± 1: The very next day Giovanni La Barbera found the new record primes: 1693965 · 266443 ± 1: The fact that the size of these records are close is no coincidence! Before we seek a record like this, we usually try to estimate how long the search might take, and use this information to determine our search parameters. To do this we need to know how common twin primes are. It has been conjectured that the number of twin primes less than or equal to N is asymptotic to Z N dx 2C2N 2C2 2 ∼ 2 2 (log x) (log N) where C2, called the twin prime constant, is approximately 0:6601618. Using this we can estimate how many numbers we will need to try before we find a prime. In the case of Underbakke and La Barbera, they were both using the same sieving software (NewPGen1 by Paul Jobling) and the same primality proving software (Proth.exe2 by Yves Gallot) on similar hardware{so of course they choose similar ranges to search. But where does this conjecture come from? In this chapter we will discuss a general method to form conjectures similar to the twin prime conjecture above. We will then apply it to a number of different forms of primes such as Sophie Germain primes, primes in arithmetic progressions, primorial primes and even the Goldbach conjecture. -
New Congruences and Finite Difference Equations For
New Congruences and Finite Difference Equations for Generalized Factorial Functions Maxie D. Schmidt University of Washington Department of Mathematics Padelford Hall Seattle, WA 98195 USA [email protected] Abstract th We use the rationality of the generalized h convergent functions, Convh(α, R; z), to the infinite J-fraction expansions enumerating the generalized factorial product se- quences, pn(α, R)= R(R + α) · · · (R + (n − 1)α), defined in the references to construct new congruences and h-order finite difference equations for generalized factorial func- tions modulo hαt for any primes or odd integers h ≥ 2 and integers 0 ≤ t ≤ h. Special cases of the results we consider within the article include applications to new congru- ences and exact formulas for the α-factorial functions, n!(α). Applications of the new results we consider within the article include new finite sums for the α-factorial func- tions, restatements of classical necessary and sufficient conditions of the primality of special integer subsequences and tuples, and new finite sums for the single and double factorial functions modulo integers h ≥ 2. 1 Notation and other conventions in the article 1.1 Notation and special sequences arXiv:1701.04741v1 [math.CO] 17 Jan 2017 Most of the conventions in the article are consistent with the notation employed within the Concrete Mathematics reference, and the conventions defined in the introduction to the first articles [11, 12]. These conventions include the following particular notational variants: ◮ Extraction of formal power series coefficients. The special notation for formal n k power series coefficient extraction, [z ] k fkz :7→ fn; ◮ Iverson’s convention.